Probability Essentials

1. Front Matter

   Pages I-X

   PDF

2. Introduction

   • Jean Jacod, Philip Protter

   Pages 1-5

3. Axioms of Probability

   • Jean Jacod, Philip Protter

   Pages 7-13

4. Conditional Probability and Independence

   • Jean Jacod, Philip Protter

   Pages 15-20

5. Probabilities on a Finite or Countable Space

   • Jean Jacod, Philip Protter

   Pages 21-26

6. Random Variables on a Countable Space

   • Jean Jacod, Philip Protter

   Pages 27-34

7. Construction of a Probability Measure

   • Jean Jacod, Philip Protter

   Pages 35-38

8. Construction of a Probability Measure on R

   • Jean Jacod, Philip Protter

   Pages 39-46

9. Random Variables

   • Jean Jacod, Philip Protter

   Pages 47-50

10. Integration with Respect to a Probability Measure

    • Jean Jacod, Philip Protter

    Pages 51-64

11. Independent Random Variables

    • Jean Jacod, Philip Protter

    Pages 65-75

12. Probability Distributions on R

    • Jean Jacod, Philip Protter

    Pages 77-85

13. Probability Distributions on R n

    • Jean Jacod, Philip Protter

    Pages 87-101

14. Characteristic Functions

    • Jean Jacod, Philip Protter

    Pages 103-109

15. Properties of Characteristic Functions

    • Jean Jacod, Philip Protter

    Pages 111-116

16. Sums of Independent Random Variables

    • Jean Jacod, Philip Protter

    Pages 117-123

17. Gaussian Random Variables (The Normal and the Multivariate Normal Distributions)

    • Jean Jacod, Philip Protter

    Pages 125-139

18. Convergence of Random Variables

    • Jean Jacod, Philip Protter

    Pages 141-149

19. Weak Convergence

    • Jean Jacod, Philip Protter

    Pages 151-166

20. Weak Convergence and Characteristic Functions

    • Jean Jacod, Philip Protter

    Pages 167-171

We have made small changes throughout the book, including the exercises, and we have tried to correct if not all, then at least most of the typos. We wish to thank the many colleagues and students who have commented c- structively on the book since its publication two years ago, and in particular Professors Valentin Petrov, Esko Valkeila, Volker Priebe, and Frank Knight. Jean Jacod, Paris Philip Protter, Ithaca March, 2002 Preface to the Second Printing of the Second Edition We have bene?ted greatly from the long list of typos and small suggestions sent to us by Professor Luis Tenorio. These corrections have improved the book in subtle yet important ways, and the authors are most grateful to him. Jean Jacod, Paris Philip Protter, Ithaca January, 2004 Preface to the First Edition We present here a one semester course on Probability Theory. We also treat measure theory and Lebesgue integration, concentrating on those aspects which are especially germane to the study of Probability Theory. The book is intended to ?ll a current need: there are mathematically sophisticated s- dents and researchers (especially in Engineering, Economics, and Statistics) who need a proper grounding in Probability in order to pursue their primary interests. Many Probability texts available today are celebrations of Pr- ability Theory, containing treatments of fascinating topics to be sure, but nevertheless they make it di?cult to construct a lean one semester course that covers (what we believe) are the essential topics.

• Measure
• Normal distribution
• Probability distribution
• Probability theory
• Random variable
• operations research
• quantitative finance

"(The book is) a lean and largely self-contained introduction to the modern theory of probability, aimed at advanced undergraduate or beginning graduate students. The 28 short chapters belie the book's genesis as polished lecture notes; the exposition is sleek and rigorous and each chapter ends with a supporting collection of mainly routine exercises. ... The authors make it clear what luggage is required for this exhilarating trek,... a good knowledge of advanced calculus, some linear algebra, and some "mathematical sophistication". With this understood, the itinerary is immaculately paced and planned with just the right balances of technical ascents and pauses to admire the scenery. Within the constraints of a slim volume, it is hard to imagine how the authors could have done a more effective or more attractive job." The Mathematical Gazette, Vol. 84, No 500, 2000 "The authors provide the shortest path through the twenty-eight chapter headings. The topics are treated in a mathematically and pedagogically digestible way. The writing is concise and crisp: the average chapter length is about eight pages. ... Numerous exercises add to the value of the text as a teaching tool. In conclusion, this is an excellent text for the intended audience."
Short Book Reviews, Vol. 21, No. 2, 2001

• Laboratoire de Probabilités, Université de Paris, Paris Cedex 05, France

  Jean Jacod

• School of Operations Research and Industrial Engineering, Cornell University, Ithaca, USA

  Philip Protter